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Theorem difdif 3959
 Description: Double class difference. Exercise 11 of [TakeutiZaring] p. 22. (Contributed by NM, 17-May-1998.)
Assertion
Ref Expression
difdif (𝐴 ∖ (𝐵𝐴)) = 𝐴

Proof of Theorem difdif
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 pm4.45im 819 . . 3 (𝑥𝐴 ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐴)))
2 iman 392 . . . . 5 ((𝑥𝐵𝑥𝐴) ↔ ¬ (𝑥𝐵 ∧ ¬ 𝑥𝐴))
3 eldif 3802 . . . . 5 (𝑥 ∈ (𝐵𝐴) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝐴))
42, 3xchbinxr 327 . . . 4 ((𝑥𝐵𝑥𝐴) ↔ ¬ 𝑥 ∈ (𝐵𝐴))
54anbi2i 616 . . 3 ((𝑥𝐴 ∧ (𝑥𝐵𝑥𝐴)) ↔ (𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐵𝐴)))
61, 5bitr2i 268 . 2 ((𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐵𝐴)) ↔ 𝑥𝐴)
76difeqri 3953 1 (𝐴 ∖ (𝐵𝐴)) = 𝐴
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2107   ∖ cdif 3789 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-v 3400  df-dif 3795 This theorem is referenced by:  dif0  4181  undifabs  4269
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